Question: Perform the row operation, $R_2-2R_1\rightarrow R_2$, on the following matrix. $\left[\begin{array} {ccc} -2 & -1 & 2 & 1 \\ 1 & 1 & -5 & 0 \\ 1 & 1 & -5 & 0 \end{array} \right] $
Explanation: Background There are three basic row operations that can be performed on matrices. $R_i \leftrightarrow R_j$. This symbol tells us to interchange rows $i$ and $j$. $cR_i \rightarrow R_i$. This symbol tells us to multiply a row $i$ by a constant $c$. $R_i + cR_j \rightarrow R_i$. This symbol tells us to add $c$ times row $j$ to row $i$. Finding the new row to be used For the given matrix, $R_1$ and $R_2$ are given below. $R_1=\left[\begin{array} {ccc} -2 & -1 & 2 & 1 \end{array} \right] ~~~~~ R_2=\left[\begin{array} {ccc} 1 & 1 & -5 & 0 \end{array} \right]$ We are asked to perform the row operation, $R_2-2R_1\rightarrow R_2$. Therefore, we must add $-2R_1$ to $R_2$. $\begin{aligned}R_2-2R_1 &= \left[\begin{array} {ccc} 1 & 1 & -5 & 0 \end{array} \right] - 2\left[\begin{array} {ccc} -2 & -1 & 2 & 1 \end{array} \right] \\\\&=\left[\begin{array} {ccc} 5 & 3 & -9 & -2 \end{array} \right]\end{aligned}$ Substituting the row Now, we must substitute row $R_2$ with $R_2-2R_1$. $\left[\begin{array} {ccc} -2 & -1 & 2 & 1 \\ {1} & {1} & {-5} & {0} \\ 1 & 1 & -5 & 0 \end{array} \right] \xrightarrow{R_2-2R_1 \rightarrow R_2} \left[\begin{array} {ccc} -2 & -1 & 2 & 1 \\ {5} & {3} & {-9} & {-2 } \\ 1 & 1 & -5 & 0 \end{array} \right]$ Summary Our resultant matrix is the following. $\left[\begin{array} {ccc} -2 & -1 & 2 & 1 \\ 5 & 3 & -9 & -2 \\ 1 & 1 & -5 & 0 \end{array} \right]$